Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
16.3-a1 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{15} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$36.37942472$ |
1.202010026 |
\( \frac{180079}{8} a^{2} - \frac{324541}{8} a - \frac{108845}{8} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} + a - 3\) , \( a + 1\) , \( a^{2} + a - 3\) , \( a^{2} - 4\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(a^{2}+a-3\right){x}+a^{2}-4$ |
16.3-a2 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{36} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$18.18971236$ |
1.202010026 |
\( -\frac{4350984313}{16777216} a^{2} + \frac{2200640379}{16777216} a + \frac{14807078979}{16777216} \) |
\( \bigl[a^{2} - 3\) , \( -a + 1\) , \( a + 1\) , \( -3 a^{2} + 2 a + 7\) , \( 3 a^{2} - 8 a\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a^{2}+2a+7\right){x}+3a^{2}-8a$ |
16.3-a3 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{15} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$18.18971236$ |
1.202010026 |
\( -\frac{83810347122767}{8} a^{2} + \frac{21296354689181}{8} a + \frac{329829957244973}{8} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 2\) , \( a^{2} - 3\) , \( 54 a^{2} - 92 a - 45\) , \( 34 a^{2} - 43 a - 60\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(54a^{2}-92a-45\right){x}+34a^{2}-43a-60$ |
16.3-a4 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$72.75884944$ |
1.202010026 |
\( -\frac{106156153}{64} a^{2} + \frac{26982267}{64} a + \frac{417901187}{64} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 2\) , \( a^{2} - 3\) , \( -11 a^{2} + 23 a + 5\) , \( 24 a^{2} - 42 a - 15\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-11a^{2}+23a+5\right){x}+24a^{2}-42a-15$ |
16.3-a5 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$36.37942472$ |
1.202010026 |
\( \frac{3845384044025}{64} a^{2} - \frac{7077548261243}{64} a - \frac{2046702860291}{64} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( 49 a^{2} - 26 a - 228\) , \( -420 a^{2} + 6 a + 1483\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(49a^{2}-26a-228\right){x}-420a^{2}+6a+1483$ |
16.3-a6 |
16.3-a |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{24} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$72.75884944$ |
1.202010026 |
\( \frac{416808455}{4096} a^{2} - \frac{687196165}{4096} a - \frac{187854653}{4096} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( 9 a^{2} - a - 33\) , \( 6 a^{2} - 3 a - 26\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(9a^{2}-a-33\right){x}+6a^{2}-3a-26$ |
16.3-b1 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$4$ |
\( 2 \) |
$1$ |
$5.340082704$ |
0.705765195 |
\( \frac{3845384044025}{64} a^{2} - \frac{7077548261243}{64} a - \frac{2046702860291}{64} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 2\) , \( a^{2} - 3\) , \( -37 a^{2} - 63 a - 26\) , \( -201 a^{2} - 393 a - 127\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-37a^{2}-63a-26\right){x}-201a^{2}-393a-127$ |
16.3-b2 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{24} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$42.72066163$ |
0.705765195 |
\( \frac{416808455}{4096} a^{2} - \frac{687196165}{4096} a - \frac{187854653}{4096} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 2\) , \( a^{2} - 3\) , \( -2 a^{2} - 3 a - 1\) , \( -a^{2} - 2 a - 2\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-2a^{2}-3a-1\right){x}-a^{2}-2a-2$ |
16.3-b3 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{15} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$170.8826465$ |
0.705765195 |
\( \frac{180079}{8} a^{2} - \frac{324541}{8} a - \frac{108845}{8} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - 4\) , \( a^{2} - 3\) , \( 4 a\) , \( 3 a^{2} - a - 3\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+4a{x}+3a^{2}-a-3$ |
16.3-b4 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{36} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$10.68016540$ |
0.705765195 |
\( -\frac{4350984313}{16777216} a^{2} + \frac{2200640379}{16777216} a + \frac{14807078979}{16777216} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 4\) , \( a + 1\) , \( -8 a^{2} + 11 a + 13\) , \( -38 a^{2} + 65 a + 24\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-8a^{2}+11a+13\right){x}-38a^{2}+65a+24$ |
16.3-b5 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{15} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$85.44132327$ |
0.705765195 |
\( -\frac{83810347122767}{8} a^{2} + \frac{21296354689181}{8} a + \frac{329829957244973}{8} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a + 1\) , \( 166 a^{2} - 316 a - 91\) , \( -552 a^{2} + 1033 a + 298\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(166a^{2}-316a-91\right){x}-552a^{2}+1033a+298$ |
16.3-b6 |
16.3-b |
$6$ |
$8$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$2.14656$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$170.8826465$ |
0.705765195 |
\( -\frac{106156153}{64} a^{2} + \frac{26982267}{64} a + \frac{417901187}{64} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a + 1\) , \( -44 a^{2} + 79 a + 24\) , \( -43 a^{2} + 79 a + 23\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-44a^{2}+79a+24\right){x}-43a^{2}+79a+23$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.